Dispersive blow-up for the fifth order Korteweg-de Vries equation on the line

Abstract

In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R,\; t>0,\\ u(x,0)&\hspace{-2mm}=u_0(x),& \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s,b}$ for appropriate values of $s$ and $b$, along with a regularity property for the nonlinear part of that solution. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon.